L intersection bodies - COnnecting REpositoriesintersection bodies with respect to the usual radial topology on Sn.TheLp analogue of a result of Hensley on intersection bodies will

Advances in Mathematics 217 (2008) 2599–2624www.elsevier.com/locate/aim

Lp intersection bodies

Christoph Haberl 1

Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstraße 8-10,1040 Vienna, Austria

Received 13 July 2007; accepted 13 November 2007

Available online 21 December 2007

Communicated by Michael J. Hopkins

Abstract

Basic relations and analogies between intersection bodies and their symmetric and nonsymmetric Lp

counterparts are established.© 2007 Elsevier Inc. All rights reserved.

Keywords: Intersection body; Dual Brunn–Minkowski theory

1. Introduction

The celebrated Busemann–Petty problem asks the following: if K and L are origin-symmetricn-dimensional convex bodies such that all (n − 1)-dimensional volumes of central hyperplanesections of K are less than the corresponding sections of L, does it follow that the volume of K

is less than the volume of L? It turned out that the answer is affirmative for n � 4 and negativefor n > 4 (see, e.g., [4,8,41]). Intersection bodies, which were introduced by Lutwak [25], playeda crucial role for the solution of this problem. These bodies are also fundamental in geometrictomography (see, e.g., [5]), in affine isoperimetric inequalities (see, e.g., [38]) and the geome-try of Banach spaces (see, e.g., [22,39]). To give a precise definition of intersection bodies weintroduce some notation.

E-mail address: [email protected] Supported by FWF Project P18308 and the European Network MRTN-CT-2004-511953.

0001-8708/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.aim.2007.11.013

2600 C. Haberl / Advances in Mathematics 217 (2008) 2599–2624

We write ρ(K,u) := max{r � 0: ru ∈ K}, u ∈ Sn−1, for the radial function of a compactsubset K in Euclidean n-space R

n which is starshaped with respect to the origin. If ρ(K, ·) iscontinuous, such a set K is called star body. Let Sn denote the set of star bodies in R

n. Theintersection body operator assigns to each K ∈ Sn the star body IK with radial function

ρ(IK,u) = vol(K ∩ u⊥)

, u ∈ Sn−1,

where vol denotes (n − 1)-dimensional volume and u⊥ is the hyperplane orthogonal to u.Ludwig [24] characterized the intersection body operator by its compatibility with linear

maps and its valuation property. She proved that the intersection body operator is the only non-trivial GL(n) contravariant L1 radial valuation. This result is part of the dual Brunn–Minkowskitheory. The corresponding characterization within the dual Lp Brunn–Minkowski theory (see,e.g., [6,32] for other recent contributions to this theory) was established in [16]. It turned out,that for Lp radial valuations one has to distinguish between valuations having centrally symmet-ric images or not. This phenomenon does not occur in the L1 situation. The symmetric case ofthe Lp classification result showed that the natural definition for (symmetric) Lp intersectionbodies comes from an operator Ip . For 0 < p < 1, the latter maps each K ∈ Sn to the star bodyIpK with radial function

ρ(IpK,u)p = 1

�(1 − p)

∫K

|x · u|−p dx, u ∈ Sn−1,

where � denotes the Gamma function, x · u is the usual inner product of x,u ∈ Rn, and integra-

tion is with respect to Lebesgue measure. Up to normalization, Ip equals the polar L−p centroidbody. Centroid bodies were introduced by Petty in 1961. Lutwak and Zhang [31] extended thisconcept to Lq centroid bodies for q > 1. Gardner and Giannopoulos [7] as well as Yaskin andYaskina [40] investigated extensions of this notion also for −1 < q < 1. Lq centroid bodiesthemselves were studied by many different authors (see e.g. [2,3,16,20,23,26,29,31,33,40]). Fur-thermore, they are extremely useful tools in different situations. Among others, they led Lutwak,Yang and Zhang [30] to information theoretic inequalities, and Paouris [34] used them to proveresults concerning concentration of mass for isotropic convex bodies.

In addition to the characterization mentioned before, there are further indications that Ipcan be viewed as the Lp analogue of the intersection body operator. In the solution of the Lp

Busemann–Petty problem in [40] as well as in [20] where the authors established an Lp ana-logue of an approximation result by Goodey and Weil [10] for intersection bodies, it turned outthat the Lp intersection body behaves in the Lp context like the intersection body in the dualBrunn–Minkowski theory. See also [19] for further results.

In this paper, on the one hand, we further confirm the place of Ip within the dual Lp Brunn–Minkowski theory. We prove that every intersection body of a convex body is the limit of Lp

intersection bodies with respect to the usual radial topology on Sn. The Lp analogue of a resultof Hensley on intersection bodies will be established. We prove injectivity results along withtheir stability versions for Ip which bear a strong resemblance to results for intersection bodies.Moreover, results for intersection bodies are obtained as corollaries from our considerations oftheir Lp analogues.

C. Haberl / Advances in Mathematics 217 (2008) 2599–2624 2601

On the other hand, we investigate the operator I+p . For 0 < p < 1 and K ∈ Sn, it is defined by

ρ(I+p K,u

)p = 1

�(1 − p)

∫K∩u+

|u · x|−p dx, u ∈ Sn−1,

where u+ = {x ∈ Rn: u · x � 0}. The relation

IpK = I+p K +p I−p K (1)

with I−p K := I+p (−K), provides a strong connection of these operators to (symmetric) Lp in-tersection bodies. (For a precise definition of this addition we refer to Section 2.) Moreover,I+p essentially spans the set of all GL(n) contravariant Lp radial valuations on convex polytopes.This is the nonsymmetric case of the classification result mentioned above. Thus, comparingLudwig’s characterization of the intersection body operator, I+p itself serves as a candidate for anonsymmetric Lp analogue of the operator I. We remark that I+p is closely related to the gener-alized Minkowski–Funk transform (see, e.g., [37]). The operator I+p is of considerable interestsince it is, as we will see, injective on nonsymmetric bodies. This is in contrast to other impor-tant operators in convex geometry. Most of them, like the intersection body operator, are injectiveonly on centrally symmetric sets.

Finally, we consider a nonsymmetric version of the Lp Busemann–Petty problem. In contrastto the original Busemann–Petty problem and its Lp analogue, we obtain a sufficient conditionin terms of nonsymmetric Lp intersection bodies which allows to compare volumes of nonsym-metric bodies.

2. Notation and preliminaries

We work in Euclidean n-space Rn and write x · u for the usual inner product of two vectors

x,u ∈ Rn. The Euclidean unit ball in R

n is denoted by Bn and we write Sn−1 for its boundary.The volume κn of Bn and the surface area ωn of Bn are given by

κn = πn/2

�(1 + n/2), ωn = 2πn/2

�(n/2). (2)

By a convex body we mean a nonempty, compact, convex subset of Rn. We write Kn for the set

of convex bodies in Rn and Kn

0 ⊂ Kn for the subset of convex bodies which contain the originin their interiors. For 0 < r < R, we denote by Kn(r,R) the set of convex bodies in R

n whichcontain a Euclidean ball of radius r and center at the origin and are contained in a Euclideanball with radius R and center at the origin. h(K, ·) :Sn−1 → R denotes the support function ofK ∈ Kn, i.e. h(K,u) := max{u · x: x ∈ K}. For K ∈Kn

0 , the polar body K∗ ∈ Kn0 is defined by

K∗ := {x ∈ R

n: x · y � 1 for every y ∈ K}.

Note that

ρ(K∗, ·) = 1

(3)
h(K, ·)


for every K ∈ Kn0 . Kn is topologized as usual by the topology induced from the Hausdorff dis-

tance

δ(K,L) = supu∈Sn−1

∣∣h(K,u) − h(L,u)∣∣ =: ∥∥h(K, ·) − h(L, ·)∥∥∞,

for K,L ∈Kn. The natural metric on Sn is the radial metric defined by

δ(K,L) = ∥∥ρ(K, ·) − ρ(L, ·)∥∥∞,

for K,L ∈ Sn. Occasionally, we deal with another metric on Sn which comes from the L2 normon the space of continuous functions on the sphere:

δ2(K,L) = ∥∥ρ(K, ·) − ρ(L, ·)∥∥2.

General references on star and convex bodies are [5] and [38].In the nineties, Lutwak [28] extended the classical Brunn–Minkowski theory to the Lp Brunn–

Minkowski theory. The starting point of his studies was the Lp mixed volume. For p � 1, let

Vp(K,L) = 1

n

∫Sn−1

h(L,u)ph(K,u)1−p dS(K,u), (4)

where K,L ∈Kn0 and S(K, ·) denotes the surface area measure of K . Lutwak proved in [28] that

Vp(K,L) = p

nlim

ε→0+V (K +p ε1/pL) − V (K)

ε, (5)

where h(K +p L, ·)p := h(K, ·)p +h(L, ·)p defines Lp Minkowski addition. The correspondingnotion within the dual Lp Brunn–Minkowski theory is the following. Denote by Sn

0 the set ofstar bodies containing the origin in their interiors. For K,L ∈ Sn

0 and arbitrary p ∈ R we define

Vp(K,L) := 1

n

∫Sn−1

ρ(L,u)pρ(K,u)n−p du.

For 0 < p < n, this definition extends to all elements of Sn. As before, this quantity follows frommerging volume with a certain addition, namely radial Lp addition. For p = 0, the latter assignsto two star bodies K,L ∈ Sn

0 and positive reals α,β the star body α · K +p β · L with radialfunction

ρ(α · K +p β · L, ·)p = αρ(K, ·)p + βρ(L, ·)p.

For positive p, this definition extends to all elements of Sn. By the polar formula for volume weget

Vp(K,L) = plim+

V (K +p ε · L) − V (K), (6)

n ε→0 ε


for two star bodies K,L ∈ Sn0 . For 0 < p < 1, Hölder’s inequality and the polar formula for

volume gives the dual Ln−p Minkowski and the dual Lp Minkowski inequality

Vn−p(K,L)n � V (K)pV (L)n−p, (7)

Vp(K,L)n � V (K)n−pV (L)p. (8)

If K,L = {0}, equality holds in (7) or (8) if and only if K and L are dilates. The polar formulafor volume of star bodies together with the linearity properties of dual mixed volumes give

V (K +n−p L) = Vn−p(K +n−p L,K +n−p L)

= Vn−p(K +n−p L,K) + Vn−p(K +n−p L,L).

Thus (7) yields the dual Lp Kneser–Süss inequality

V (K +n−p L)(n−p)/n � V (K)(n−p)/n + V (L)(n−p)/n. (9)

Equality holds for star bodies K,L ∈ Sn, K,L = {0}, if and only if they are dilates.For p < 1, p = 0, and functions f ∈ C(Sn−1), the L−p cosine transform is defined by

C−pf (v) =∫

Sn−1

|u · v|−pf (u)du, v ∈ Sn−1.

We further introduce the nonsymmetric L−p cosine transform

C+−pf (v) =∫

Sn−1∩v+

|u · v|−pf (u)du, v ∈ Sn−1.

Note that a change into polar coordinates proves

ρ(IpK,v)p = ((n − p)�(1 − p)

)−1C−pρ(K, ·)n−p(v), (10)

ρ(I+p K,v

)p = ((n − p)�(1 − p)

)−1C+−pρ(K, ·)n−p(v), (11)

for every v ∈ Sn−1. This enables us to show that Ip and I+p map Bn to balls of radii rIp and rI+p ,respectively. Indeed, relation (11) yields

ρ(I+p Bn, v

)p = ωn−1

(n − p)�(1 − p)

1∫0

t−p(1 − t2)(n−3)/2

dt

= ωn−1�((1 − p)/2)�((n − 1)/2).

2(n − p)�(1 − p)�((n − p)/2)


So by (2) and the formula

�(2x) = 22x−1

√π

�(x)�

(x + 1

2

), (12)

which holds for complex numbers x and x + 12 that do not belong to −N ∪ {0}, we obtain

rp

I+p= 2pπn/2

(n − p)�((n − p)/2)�(1 − p/2)(13)

for p < 1. Obviously, rpIp

= 2rp

I+p.

3. Relations between intersection bodies and their Lp analogues

Our first theorem clarifies the behavior of the Lp intersection body of a convex body K as p

tends to one.

Theorem 1. For every K ∈Kn0 , we have

δ(I±p K, IK

) → 0 and δ(IpK,2IK) → 0,

for p ↗ 1.

(Compare [22, p. 9] and [7, Proposition 3.1].) In [16] it was shown that the operators I+p and I−pessentially span the set of nontrivial GL(n) covariant Lp radial valuations on convex polytopesfor 0 < p < 1. But the intersection body operator I is the only nontrivial L1 radial valuation(see [24]). So in some sense Theorem 1 explains the surprising fact that the set of Lp radialvaluations is two-parametric for 0 < p < 1 and only one-parametric for p = 1.

Before we start to prove this approximation result, we remark that the radial function of I+pcan be given in terms of fractional derivatives. Suppose h is a continuous, integrable function onR that is m-times continuously differentiable in some neighborhood of zero. For −1 < q < m,q = 0,1, . . . ,m − 1, the fractional derivative of order q of the function h at zero is defined as

h(q)(0) = 1

�(−q)

1∫0

t−1−q

(h(t) − h(0) − · · · − h(m−1)(0)

tm−1

(m − 1)!)

dt

+ 1

�(−q)

∞∫1

t−1−qh(t) dt + 1

�(−q)

m−1∑k=0

h(k)(0)

k!(k − q).

For a non-negative integer k < m we have

lim h(q)(0) = (−1)kdk

kh(t)

∣∣∣∣ . (14)
q→k dt t=0


For 0 < p < 1 and K ∈ Kn0 Fubini’s theorem gives

ρ(I+p K,v

)p = 1

�(1 − p)

∞∫0

t−pAK,v(t) dt = A(p−1)K,v (0), (15)

where AK,v(t) := vol(K ∩ {x ∈ Rn: x · v = t}) denotes the parallel section function of K in

direction v ∈ Sn−1. For details on fractional derivatives we refer to [22, Section 2.6].

Proof. Suppose 0 < p < 1. First, we prove the pointwise convergence

ρ(I+p K,u

) → ρ(IK,u), u ∈ Sn−1, (16)

as p tends to one (compare [22, p. 9], [7, Proposition 3.1]). We can approximate K ∈ Kn0 with

respect to the Hausdorff metric by bodies belonging to Kn0 which have infinitely smooth sup-

port functions (see, e.g., [38, Theorem 3.3.1]). By (3), this yields an approximation of K withrespect to the radial metric by convex bodies with infinitely smooth radial functions. Note thatby (13) and the representation of the radial function of I as spherical Radon transform (see [25,formula 8.5]) we obtain for K1,K2 ∈Kn(r,R) with R > 1, p > 1/2∣∣ρ(

I+p K1, u) − ρ

(I+p K2, u

)∣∣ � c1(n)R2nδ(K1,K2),∣∣ρ(IK1, u) − ρ(IK2, u)∣∣ � c2(n)Rn−2δ(K1,K2),

where c1(n), c2(n) are constants depending on n only. So in order to derive (16), we can restrictourselves to bodies K ∈ Kn

0 with sufficiently smooth radial functions. For such bodies, AK,u iscontinuously differentiable in a neighbourhood of 0 (cf. [22, Lemma 2.4]). Thus (14) and (15)prove (16).

For k ∈ N, let 0 < pk < 1 be an increasing sequence which converges to one. Define functions

f 1k (u) := ρ

(I+pk

K,u)−1

(�(1 + n)V (K ∩ u+)

�(1 − pk + n)

)1/pk

,

f 2k (u) :=

(�(1 − pk + n)

�(1 + n)

)1/pk

,

f 3k (u) := V

(K ∩ u+)−1/pk ,

on Sn−1. We need the following result: For a compact convex set K with nonempty interior anda concave function f :K → R

+, the function

F(q) :=(

1

nB(q + 1, n)V (K)

∫K

f (x)q dx

)1/q

,

where B denotes the beta function, is decreasing on (−1,0) (see [9] and references therein).Thus the sequence f 1 is increasing.
k


Since o is an interior point of K , there exists a constant c > 0 such that cV (K ∩ u+) � 1for every u ∈ Sn−1. Thus c−1/pkf 3

k is increasing, too. So f 1k and c−1/pkf 3

k are monotone se-quences of continuous functions converging pointwise to continuous functions on a compact set.Therefore they converge uniformly by Dini’s theorem. Thus

ρ(I+pk

K,u)−1 = f 1

k f 2k f 3

k (u) → ρ(IK,u)−1

uniformly for k → ∞.The other assertions of the theorem immediately follow from the definition I−p K = I+p (−K)

and relation (1). �Next, we prove an inequality between radial functions of intersection bodies and their Lp

analogues.

Theorem 2. Suppose 0 < p < 1. For all symmetric K ∈ Kn0 with volume one there exist positive

constants c1, c2 independent of the dimension n, the body K and p, such that

c1ρ(IK,u) � ρ(IpK,u) � c2ρ(IK,u)

holds for every direction u ∈ Sn−1.

Proof. We use techniques of Milman and Pajor [33]. The following two facts can also be foundin this paper. For a measurable function f : Rn → R

+ which has values less than or equal 1 anda symmetric convex body Q ∈ Kn

0 , the function

F1(q) :=(∫

Rn ρ(Q,x)−qf (x) dx∫Q

ρ(Q,x)−q dx

)1/(n+q)

is increasing on (−n,∞).Suppose ψ : R+ → R

+ satisfies ψ(0) = 0, ψ and ψ(x)/x are increasing on an interval (0, ν],and ψ(x) = ψ(ν) for x � ν. Let h : R+ → R+ be a decreasing, continuous function which van-ishes at ψ(ν). Then

F2(q) :=(∫ ∞

0 h(ψ(x))xq dx∫ ∞0 h(x)xq dx

)1/(1+q)

is a decreasing function on (−1,∞) (provided that the integrals make sense).To prove the second inequality take f (x) := AK,u(x)/AK,u(0) and Q := [−1,1] ⊂ R.

Brunn’s theorem shows that this f satisfies the above assumptions to ensure that F1(−p) �F1(0), that is (

(1 − p)∫

R|x|−pAK,u(x) dx

2 vol(K ∩ u⊥)

)1/(1−p)

� 1

2 vol(K ∩ u⊥).

Thus by (15)

ρ(IpK,u) � 21/p

ρ(IK,u).
(�(2 − p))


We have limp→0+(�(2 − p))1/p = exp(γ − 1) > 0 where γ denotes the Euler–Mascheroni con-stant. For all other values of p ∈ (0,1] we trivially have that (�(2 − p))1/p > 0. This shows that(�(2 − p))1/p can be bounded from below on (0,1) by a positive constant smaller than one.

To establish the first inequality take h(x) = (1 − x)n−1I[0,1](x), x � 0 and ψ(x) = 1 −

(AK,u(x)/AK,u(0))1/(n−1) for arbitrary u ∈ Sn−1. (I stands for the indicator function.) Brunn’stheorem shows that ψ is a convex function on [0, h(K,u)]. Therefore these two functions satisfythe above conditions to guarantee the monotonicity of F2. Hence F2(−p) � F2(0), which can berewritten as ( ∫ ∞

0 AK,u(x)x−p dx

vol(K ∩ u⊥)B(1 − p,n)

)1/(1−p)

� n

2 vol(K ∩ u⊥).

Using (15), we obtain

ρ(IpK,u) � 2

(�(n)n1−p

�(1 + n − p)

)1/p

ρ(IK,u).

We want to show that

�(n)n1−p

�(1 + n − p)� 1

for every n ∈ N and p ∈ (0,1). So we have to prove that

ln�(n + 1 − p) + p lnn � ln�(n + 1). (17)

Since the Gamma function is logarithmic convex we get

ln�(n + 1 − p) = ln�((1 − p)(n + 1) + pn

)� (1 − p) ln�(n + 1) + p ln�(n)

= (1 − p) lnn + ln�(n).

This immediately implies (17). �Now, we give applications of Theorem 2. A compact set K ⊂ R

n with volume 1 is said to bein isotropic position if for each unit vector u∫

K

(x · u)2 = L2K.

LK is called isotropic constant of K . Let K ∈ Kn0 be symmetric and in isotropic position. Hens-

ley [17] proved the existence of absolute (not depending on K and n) constants c1, c2 such that

c1 � ρ(IK,u) � c2, ∀u,v ∈ Sn−1.
ρ(IK,v)


In fact, even more is true, namely

c1

LK

� ρ(IK,u) � c2

LK

(18)

for all unit vectors u and universal constants c1, c2.Hensley’s original relation combined with Theorem 2 gives the Lp analogue of Hensley’s

result.

Theorem 3. Assume 0 < p < 1. For symmetric bodies K ∈ Kn0 in isotropic position there exist

constants c1, c2 independent of the dimension n, the body K and p, such that

c1 � ρ(IpK,u)

ρ(IpK,v)� c2

for all u,v ∈ Sn−1.

One of the major open problems in the field of convexity is the slicing conjecture. It askswhether LK for centrally symmetric convex bodies K in isotropic position can be bounded fromabove by a universal constant. Relation (18) shows that this is equivalent to bound ‖ρ(IK, ·)−1‖∞by a constant independent of the dimension and the body K . By Theorem 2, the slicing conjectureis equivalent to ask whether there exists a constant c independent of the dimension and the bodyK such that ∥∥ρ(IpK, ·)−1

∥∥∞ � c

for all symmetric K ⊂ Kn in isotropic position and some p ∈ (0,1).

4. An Lp ellipsoid formula

Busemann showed that the volume of a centered ellipsoid E ⊂ Rn can essentially be obtained

by averaging over certain powers of (n − 1)-dimensional volumes of its hyperplane sections. Tobe precise,

V (E)n−1 = κn−2n

nκnn−1

∫Sn−1

vol(E ∩ u⊥)n

du. (19)

This formula is the hyperplane case of a more general version due to Furstenberg and Tzkoni(cf. [5, Corollary 9.4.7]). Guggenheimer [15] established a companion of (19) which involvesthe surface area of E, S(E):

V (E)n−1S(E) = κn−1n

κn+1n−1

∫n−1

vol(E ∩ u⊥)n+1

du. (20)

S


Lutwak [27] obtained a more general ellipsoid formula which contains (19) and (20) as specialcases:

κn−2n

κnn−1

∫Sn−1

vol(E ∩ u⊥)n+1

vol(F ∩ u⊥)du = V (E)n−1

V (F)

∫Sn−1

h(F,u)dS(E,u),

where E,F ⊂ Rn are centered ellipsoids. For E = Bn, this result establishes a formula similar

to (20) involving the mean width of E.We extend this formula using Lp intersection bodies. From our equation one can obtain the

formulas of Busemann, Guggenheimer, and Lutwak by taking the limit p ↗ 1.

Theorem 4. For 0 < p < 1 and two centered ellipsoids E and F we have

Vp−2(I+p E, I+p F

) = rn

I+pκ

2−n/pn V (E)(n−3p+2)/pV (F )(p−2)/pV2−p(E,F ). (21)

Proof. We denote by E, F the ellipsoids which are dilates of E, F with volume κn. Thus

E = λE and F = μF

where

λ := (κn/V (E)

)1/n and μ := (κn/V (F )

)1/n.

We write φE for the linear transformation which maps the unit ball Bn to E. So φE has determi-nant ±1. The main tool in the proof will be the equation

Vp−2(E∗, F ∗) = V2−p(E, F ). (22)

From (5) and (6) we get for φ ∈ SL(n) that

Vp−2(φK,φL) = Vp−2(K,L), V2−p(φK,φL) = V2−p(K,L).

Identity (4) shows

V2−p(K,L) = 1

n

∫Sn−1

h(L,u)2−ph(K,u)p−1 dS(K,u).

Hence

V2−p(Bn,L) = Vp−2(Bn,L

∗).These preparations enable us to derive (22) by

Vp−2(E∗, F ∗) = Vp−2

((φEBn)

∗, F ∗) = Vp−2(φ−t

EBn, F

∗) = Vp−2(Bn,φ

t

EF ∗)

= V2−p

(Bn,

(φt

EF ∗)∗) = V2−p

(Bn,φ

−1E

F) = V2−p(φEBn, F )

= V2−p(E, F ).


We use obvious homogeneity properties of Vp−2 and V2−p , which follow from their integralrepresentations, for extending (22) to our ellipsoids E and F . Indeed,

Vp−2(E∗,F ∗) = Vp−2

((λ−1E

)∗,(μ−1F

)∗) = Vp−2(λE∗,μF ∗)

= λn+2−pμp−2Vp−2(E∗, F ∗) = λn+2−pμp−2V2−p(E, F )

= λ2nV2−p(E,F ). (23)

As was shown in Section 2, I+p maps the unit ball Bn to the ball rI+p Bn. Since I+p φK = φ−t I+p K

for φ ∈ SL(n), we have

I+p E = I+p λ−1E = λ1−n/pI+p E = λ1−n/pI+p φEBn

= λ1−n/prI+p φ−t

EBn = λ1−n/prI+p E∗

= λ−n/prI+p E∗.

We obtain

Vp−2(I+p E, I+p F

) = rn

I+pλ−n/p(n+2−p)μ−n/p(p−2)Vp−2

(E∗,F ∗)

= rn

I+pλ−n/p(n+2−p)+2nμ−n/p(p−2)V2−p(E,F ).

Substituting the values of λ and μ finishes the proof. �An application of Theorem 1 to (21) for the special choice E = F proves Busemann’s for-

mula (19). Guggenheimer’s relation (20) is the limiting case p ↗ 1 for F = Bn of (21). Takingthe limit p ↗ 1 in (21) without further assumptions on the involved ellipsoids yields Lutwak’sformula for intersection bodies.

5. Injectivity results

We start by collecting some basic facts about spherical harmonics. All of them can be foundin [13].

Let {Ykj : j = 1, . . . ,N(n, k)} be an orthonormal basis of the real vector space of sphericalharmonics of order k ∈ N ∪ {0} and dimension n. We write

f ∼∞∑

k=0

Yk (24)

for the condensed harmonic expansion of a function f ∈ L2(Sn−1), where

Yk =N(n,k)∑

(f,Ykj )Ykj .

j=1


Here, (f, g) stands for the usual scalar product∫Sn−1 f (u)g(u)du on L2(S

n−1). The norminduced by this scalar product is denoted by ‖ .‖2. For a bounded integrable functionΦ : [−1,1] → R we define a transformation TΦ on C(Sn−1) by

(TΦf )(v) :=∫

Sn−1

Φ(u · v)f (u)du, v ∈ Sn−1.

If Yk is a spherical harmonic of degree k, then the Funk–Hecke Theorem states that

TΦYk = an,k(TΦ)Yk (25)

with

an,k(TΦ) = ωn−1

1∫−1

Φ(t)P nk (t)

(1 − t2)(n−3)/2

dt, (26)

where P nk is the Legendre polynomial of dimension n and degree k. If (24) holds, then

TΦf ∼∞∑

k=0

an,k(TΦ)Yk. (27)

This remains true for arbitrary Φ provided the induced transformation TΦ maps continuousfunctions to continuous functions, satisfies (TΦf,g) = (f,TΦg) for all f,g ∈ C(Sn−1) as wellas (25). So (27) and Parseval’s equality show that such transformations TΦ are injective onC(Sn−1) if all multipliers an,k(TΦ) are not equal to zero.

If m � 0, �mo stands for the m-times iterated Beltrami operator. For a function f :Sn−1 → R

for which (24) holds and �mo f exists and is continuous, we have

�mo f ∼ (−1)m

∞∑k=0

km(k + n − 2)mYk. (28)

We will deal with smooth functions on the sphere and their development into series of sphericalharmonics. For this purpose, we need information on the behavior of derivatives of sphericalharmonics. For an n-dimensional spherical harmonic Yk of order k and all u ∈ Sn−1

∣∣(DαYk

(x/‖x‖))

x=u

∣∣ � cn,|α|kn/2+|α|−1‖Yk‖2, (29)

where α = (α1, . . . , αn), Dα = ∂ |α|/(∂x1)α1 · · · (∂xn)

αn and |α| = α1 + · · · + αn. Define

cn,k,p = πn/2−1�(1 − p)�((k + p)/2)

2−p�((n + k − p)/2). (30)


Lemma 5. Assume p < 1 and that p is not an integer. Then the multipliers of C+−p and C−p are

an,k

(C+−p

) ={

(−1)k/2+1cn,k,p cos(π 1+p2 ), k even,

(−1)(k−1)/2cn,k,p sin(π1+p

2 ), k odd,

and

an,k(C−p) ={

(−1)k/2+12cn,k,p cos(π 1+p2 ), k even,

0, k odd.

The multipliers an,k(C−p) appeared in their full generality already in [21] and [36]. In oursituation they are an obvious consequence of the formula for an,k(C

+−p). In dimensions three andhigher, Rubin [37] calculated an,k(C

+−p). We present another proof and establish the representa-tion of the multipliers also in dimension two.

Proof. First, we assume that n = 2. Then the relation

P 2k (t) = cos(k arccos t), k ∈ N ∪ {0},

holds for t ∈ [−1,1]. Therefore we obtain

a2,k

(C+−p

) = 2

1∫0

t−p(1 − t2)−1/2 cos(k arccos t) dt

= 2

π/2∫0

cos−p t coskt dt

= π�(1 − p)

2−p�((2 − p + k)/2)�((2 − p − k)/2),

where the last equality follows from [35, vol. 1, 2.5.11, formula 22]. If x ∈ C is not a real integer,then Euler’s reflection formula states

�(x)�(1 − x) = π

sinπx.

Thus

π

�((2 − p − k)/2)= �

((p + k)/2

)sin

(π(p + k)/2

),

which finally gives

a2,k

(C+−p

) = �(1 − p) sin(π(k + p)/2)�((k + p)/2)

2−p�((2 + k − p)/2).

An application of a standard addition theorem to the involved sine proves the first part of thelemma in dimension two.


Now, let n � 3. Then we can use the following connection between Legendre polynomials P nk

and Gegenbauer polynomials C(n−2)/2k :

P nk (t) =

(k + n − 3

n − 3

)−1

C(n−2)/2k (t). (31)

Assume further that k = 2m + 1,m ∈ N ∪ {0}. Combining (31) and (26) we obtain

an,k

(C+−p

) = ωn−1

(k + n − 3

n − 3

)−1 1∫0

t−p(1 − t2)(n−3)/2

C(n−2)/2k (t) dt.

The odd part of [35, vol. 2, 2.21.2, formula 5] yields the following expression for the integralabove:

(−1)m22m

(2m + 1)!(

n − 2

2

)m+1

(1 + p

2

)m

B

(n − 1

2+ m,

2 − p

2

),

where (a)l denotes the Pochhammer symbol. Rewriting this in terms of Gamma functions gives

an,k

(C+−p

) = 2π(n−1)/2

�((n − 1)/2)

(k + n − 3

n − 3

)−1(−1)(k−1)/22k−1

k!× �((n + k − 1)/2)

�((n − 2)/2)

�((p + k)/2)

�((1 + p)/2)

�((n − 2 + k)/2)�((2 − p)/2)

�((n + k − p)/2). (32)

Formula (12) yields

�

(n − 2 + k

2

)�

(n − 1 + k

2

)= �(n − 2 + k)

√π

2n−3+k,

�

(n − 2

2

)�

(n − 1

2

)= �(n − 2)

√π

2n−3.

Substituting this in relation (32) one obtains

an,k

(C+−p

) = π(n−1)/2(−1)(k−1)/2�((k + p)/2)�((2 − p)/2)

�((1 + p)/2)�((n + k − p)/2).

Since

�

(1 + p

2

)= π

�((1 − p)/2) sin(π(1 + p)/2),

�

(1 − p

2

)�

(2 − p

2

)=

√π�(1 − p)

2−p,

we obtain the desired representation of an,k(C+−p) in the odd case.


If k is even, one can proceed in a similar way by using the even case of [35, vol. 2, for-mula 2.21.2, 5]. The computation of the multipliers of C−p is an easy consequence of the resultsabove since Legendre polynomials of even degree are even and of odd degree are odd. �

An immediate consequence of Lemma 5 and the remarks before it is

Theorem 6. If p < 1 is not an integer, then the transformations C+−p :C(Sn−1) → C(Sn−1) and

C−p :Ce(Sn−1) → Ce(S

n−1) are injective.

(Ce(Sn−1) stands for continuous, even functions on the sphere.) The representations of the mul-

tipliers an,k(C+−p) and an,k(C−p) obtained in Lemma 5 allow us to extend them to all p ∈ R\Z.

For 0 < p < 1, there exist constants c1, c2 by Stirling’s formula which depend only on n suchthat for sufficiently large k

∣∣an,k

(C+−p

)−1∣∣ �{

c1|cos(π 1+p2 )|−1�(1 − p)−1kβ, k even,

c2|sin(π1+p

2 )|−1�(1 − p)−1kβ, k odd,(33)

where β = n/2 − p.For f ∈ C∞(Sn−1) which satisfies (24) we set for arbitrary p ∈ R\Z

C+−pf (u) :=∞∑

k=0

an,k

(C+−p

)Yk(u), for u ∈ Sn−1. (34)

From (28), (29), and the behavior of |an,k(C+−p)| as k becomes large, it follows that C+−p(f ) is

infinitely smooth.Let C∞

e (Sn−1) and C∞o (Sn−1) denote the subspaces of even and odd infinitely smooth

functions on the sphere, respectively. Denote by πe , πo the projections which assign to eachf ∈ C∞(Sn−1) its even part (f (u) + f (−u))/2 and odd part (f (u) − f (−u))/2, respectively.Define

c−1e := 2nπn−2�(1 − p)�(1 − n + p) cos

(π(1 + p)/2

)cos

(π(1 + n − p)/2

),

c−1o := 2nπn−2�(1 − p)�(1 − n + p) sin

(π(1 + p)/2

)sin

(π(1 + n − p)/2

).

The terms which involve gamma functions with a dependence on k and p in the representationsof the multipliers an,k(C

+−p) reverse if one replaces p by n − p. (This observation was usedby Koldobsky [21] for the affirmative part of the solution of the Busemann–Petty problem.)Therefore we obtain the following

Theorem 7. If p is not an integer, the transformation C+p is a bijection of C∞(Sn−1). Moreover,

the inversion formula (C+−p

)−1 = C+p−n ◦ (ceπe + coπo)

holds.


For n � 3 this was shown by Rubin [37] and the inversion formula for C−p can be foundin [36].

Now, we return to geometry. The geometric reformulation of Theorem 6 is as follows.

Theorem 8. For 0 < p < 1, the operators I±p :Sn → Sn and Ip :Sne → Sn

e are injective.

(Sne denotes the set of symmetric star bodies in R

n.) We point out that the nonsymmetric Lp

intersection body operator I+p determines also nonsymmetric star bodies uniquely. This is incontrast to its classical analogue which is injective only on centrally symmetric sets. Note thatresults by Groemer [14] and Goodey and Weil [11] ensure that certain sections determine alsoa nonsymmetric body uniquely. But in the Lp theory, the nonsymmetric Lp intersection bodyoperator is itself injective on all star bodies.

A stability version of Theorem 8 is as follows.

Theorem 9. Suppose 0 < p < 1. For γ ∈ (0,1/(1 + β)) and K,L ∈Kn(r,R) there is a constantc1 depending only on r,R,p,n, γ such that

δ(K,L) � c1δ(I+p K, I+p L

)2γ /(n+1).

If in addition K and L are symmetric, then

δ(K,L) � c2δ(IpK, IpL)2γ /(n+1),

where c2 is again a constant depending just on r,R,p,n, γ .

The proof of this result follows the approach suggested by Bourgain and Lindenstrauss [1]which was also used by Hug and Schneider [18] to establish stability results involving transfor-mations TΦ for bounded Φ .

Proof. In the proof we denote by d1, d2, . . . constants which depend on r,R,p,γ and n. Wewrite c1, c2, . . . for constants depending on r,R,n only. The ball B(0, r) is contained in K,L,hence

δ2(K,L) �((n − p)rn−p−1)−1∥∥ρ(K, ·)n−p − ρ(L, ·)n−p

∥∥2.

Groemer [12] proved that

δ(K,L) � 2

(8κn−1

n(n + 1)

)−1/(n+1)

R2r−(n+3)/(n+1)δ2(K,L)2/(n+1).

Therefore

δ(K,L) � c1((n − p)rn−p−1)−2/(n+1)∥∥ρ(K, ·)n−p − ρ(L, ·)n−p

∥∥2/(n+1)

2 . (35)

The operator I+p maps balls to balls by (13). Since I+p B(0, r) ⊂ I+p K, I+p L, we get

∥∥ρ(I+p K, ·)p − ρ

(I+p L, ·)p∥∥ � p

(rn/p−1r +

)p−1δ2

(I+p K, I+p L

).
2 Ip


Together with the trivial estimate δ2(I+p K, I+p L) � √ωnδ(I+p K, I+p L) we deduce that

∥∥ρ(I+p K, ·)p − ρ

(I+p L, ·)p∥∥

2 � c2(rn/p−1rI+p

)p−1δ(I+p K, I+p L

). (36)

So by (35) and (36) it is enough to prove∥∥ρ(K, ·)n−p − ρ(L, ·)n−p∥∥

2 � d7∥∥ρ

(I+p K, ·)p − ρ

(I+p L, ·)p∥∥γ

2 ,

for some constant d7. For simplicity we write f := ρ(K, ·)n−p − ρ(L, ·)n−p and f :=1/�(1 − p)f .

Relation (3) and the estimate∣∣h(K1, u) − h(K2, v)∣∣ � R‖u − v‖ + max

{‖u‖,‖v‖}δ(K,L)

for arbitrary vectors u,v and convex bodies K1,K2 contained in B(0, R) (cf. [38, Lemma 1.8.10])proves that f is a Lipschitz function on Sn−1 with a Lipschitz constant Λ(f ) which is at most2(n − p)Rn−p+1r−1.

Assume (24) holds for f . Since f ∈ C(Sn−1), the Poisson transform fτ satisfies

fτ (u) := 1

ωn

∫Sn−1

1 − τ 2

(1 + τ 2 − 2τ(u · v))n/2f (v) dv =

∞∑k=0

τ kYk(u),

for u ∈ Sn−1 and 0 < τ < 1 (cf. [13, Theorem 3.4.16]).Since (−β/(e ln τ))β is the maximal value of the function x → xβτx, x > 0, we have

kβτ k(1 − τ)β �(

β

−e ln τ

)β

(1 − τ)β =(

β

e

)β(1 − τ

− ln τ

)β

�(

β

e

)β

, (37)

for k ∈ N ∪ {0}. From (33) we derive the existence of a constant c3 and a positive integer N

depending on n only such that

k−β � c3 max

{∣∣∣∣cos

(π

1 + p

2

)∣∣∣∣−1

,

∣∣∣∣sin

(π

1 + p

2

)∣∣∣∣−1}· �(1 − p)−1

∣∣an,k

(C+−p

)∣∣=: c3α(p)

∣∣an,k

(C+−p

)∣∣for k � N . Define

d1 = max{

max0�k<N

{τ k(β/e)−β(1 − τ)βα(p)−1

∣∣an,k

(C+−p

)∣∣−1}, c3

}.

Thus by (37)

τ k � d1(β/e)βα(p)(1 − τ)−β∣∣an,k

(C+−p

)∣∣ =: d2(1 − τ)−β∣∣an,k

(C+−p

)∣∣


for k ∈ N ∪ {0}. Combining this with Parseval’s equation and (27) gives

‖fτ‖22 =

∞∑k=0

τ 2k‖Yk‖22 � d2

2 (1 − τ)−2β∞∑

k=0

∣∣an,k

(C+−p

)∣∣2‖Yk‖22

= d22 (1 − τ)−2β

∥∥C+−pf∥∥2

2 = d23 (1 − τ)−2β

∥∥C+−pf∥∥2

2 (38)

where d3 := �(1−p)d2. The Cauchy–Schwarz inequality, the estimate ‖f −fτ‖∞ � c4Λ(f )(1−τ) ln(2/(1 − τ)) for τ ∈ [1/4,1) (cf. [13, Lemma 5.5.8]) and (38) yield

‖f ‖22 �

∣∣(f,f − fτ )∣∣ + ∣∣(f,fτ )

∣∣ �∫ ∣∣f (u)

∣∣du‖f − fτ‖∞ + ‖f ‖2‖fτ‖2

�(√

ωn‖f − fτ‖∞ + ‖fτ‖2)‖f ‖2

�(

c5r−1Rn−p+1(1 − τ) ln

2

1 − τ+ d3(1 − τ)−β

∥∥C+−pf∥∥

2

)‖f ‖2. (39)

By (13), the quotient ‖C+p f ‖2/R

n−p can be bounded from above by c6rp

I+p. If we set

d4 := c6(4/3)1+β

ln(8/3)rp

I+p,

then

d4(1 − τ)1+β ln2

1 − τ= ‖C+−pf ‖2

Rn−p

for a certain value τ ∈ [1/4,1) if ‖C+−pf ‖2 > 0. So finally for this τ and every γ ∈ (0,1/(1+β))

we have by (39)

‖f ‖2 �(c5r

−1(n − p)Rn−p+1d−14 Rp−n + d3

)∥∥C+−pf∥∥

2(1 − τ)−β

=: d5∥∥C+−pf

∥∥2(1 − τ)−β

� R(n−p)(1−γ )d5d1−γ

4 (1 − τ)1−γ (1+β)

(ln

2

1 − τ

)1−γ ∥∥C+−pf∥∥γ

2

� R(n−p)(1−γ )d5d1−γ

4 max{(3/4)1−γ (1+β)

(ln(8/3)

)1−γ,

21−γ (1+β)((1 − γ )/

(e(1 − γ (1 + β)

)))1−γ }∥∥C+−pf∥∥γ

2

� d5d1−γ

4 d6∥∥C+−pf

∥∥γ

2 .

In conclusion we obtain

δ(K,L) � c7((n − p)rn−p−1)−2/(n+1)(

d5d1−γ

4 d6)2/(n+1)

· (c2p(rn/p−1r +

)p−1)γ /(n+1)δ(I+p K, I+p L

)2γ /(n+1).
Ip


This settles the first part of the theorem. The proof of the second part follows the same linesnoting that f is now an even function and therefore the odd coefficients in the condensed har-monic expansion of f vanish. �

Another application of Theorem 1 is the proof of a stability theorem for intersection bodies(compare Groemer’s work [12]).

Corollary 10. For γ ∈ (0,2/n) and centrally symmetric K,L ∈ Kn(r,R) there is a constant c

depending only on r,R,n, γ such that

δ(K,L) � cδ(IK, IL)2γ /(n+1).

Proof. Choose γp = 2/(n − 2p + 2) + γ − 2/n. Then the second part of Theorem 9 gives

δ(K,L) � c2(δ(IpK,2IK) + δ(2IK,2IL) + δ(2IL, IpL)

)2γp/(n+1).

The sine-term in the definition of α(p) is not involved within the centrally symmetric case.Therefore the constant c2 converges as p tends to one as one can see from the definitions ofconstants di . �

The next two results particularly show the announced analogy between intersection bodiesand their Lp analogues. A star body is called Lp intersection body if it is contained in IpSn.

Theorem 11. Suppose 0 < p < 1 and let S ∈ Sn be an Lp intersection body. Then there exists aunique centered star body Sc with IpSc = S. Moreover, this star body is characterized by havingsmaller volume than any other star body in the preimage I−1

p S.

For intersection bodies, the corresponding result was proved by Lutwak [25, Theorem 8.8].To construct the desired body of the last theorem we need the following definition. For each starbody K ∈ Sn we define a symmetric star body by

∇pK := 1

2· K +n−p

1

2· (−K).

Proof. Let S ∈ Sn be chosen such that IpS = S. The star body

Sc := ∇pS

is centrally symmetric. Representation (10) immediately shows that IpSc = S. But Ip is injectiveon centrally symmetric sets which proves the first part of the theorem.

Since (1/2) · K = (1/2)1/(n−p)K , we obtain from (9) that

V (∇pK) � V (K) (40)

with equality if and only if K is centered. If K is an arbitrary star body which is mapped to S

by Ip , then ∇pK = ∇pS. So

V (∇pS) = V (∇pK) � V (K)


with equality if and only if K is centered by (40). This establishes the second part of the theo-rem. �Theorem 12. For given star bodies K,L ∈ Sn and 0 < p < 1, the following statements areequivalent:

IpK = IpL, (41)

∇pK = ∇pL, (42)

Vp(K,M) = Vp(L,M), for each centered star body M ∈ Sn. (43)

Formally setting p = 1 and I1 = I, the corresponding equivalence (41) ⇔ (43) was establishedin [25] and (41) ⇔ (42) can be found in [5].

Proof. First, since IpK = Ip∇pK as well as IpL = Ip∇pL and Ip is injective on centrally sym-metric star bodies, (41) implies (42). Conversely, the identity ∇pK = ∇pL means

1

2ρ(K,v)n−p + 1

2ρ(−K,v)n−p = 1

2ρ(L,v)n−p + 1

2ρ(−L,v)n−p

for every v ∈ Sn−1. Therefore

1

2

∫Sn−1

|u · v|−pρ(K,v)n−p dv + 1

2

∫Sn−1

|u · v|−pρ(K,−v)n−p dv

= 1

2

∫Sn−1

|u · v|−pρ(L,v)n−p dv + 1

2

∫Sn−1

|u · v|−pρ(L,−v)n−p dv.

The invariance properties of the spherical Lebesgue measure show that (41) holds.Second, suppose that (41) holds. Thus∫

Sn−1

|u · v|−pρ(K,v)n−p dv =∫

Sn−1

|u · v|−pρ(L,v)n−p dv, ∀u ∈ Sn−1.

By Fubini’s theorem we conclude∫Sn−1

ρ(K,v)n−p

∫Sn−1

|u · v|−pf (u)dudv

=∫

Sn−1

ρ(L,v)n−p

∫Sn−1

|u · v|−pf (u)dudv,

for suitable f . The remarks after Theorem 7 show that∫n−1

ρ(K,v)n−pF (v)dv =∫n−1

ρ(L,v)n−pF (v)dv, for F ∈ C∞e

(Sn−1).

S S


An approximation argument proves that Vp(K,M) = Vp(L,M) for each centered star body M .Finally, assume that (43) holds. Define a centered star body M by

ρ(M,u)p :=∫

Sn−1

|u · v|−pf (v) dv,

where f is now a continuous, non-negative function on the sphere. Applying (43) for this spe-cial M , we get ∫

Sn−1

f (v)(ρ(IpK,v)p − ρ(IpL,v)p

)dv = 0. (44)

For arbitrary continuous functions f we can deduce (44) by writing f as the difference of itspositive and negative part. Thus ρ(IpK, ·)p = ρ(IpL, ·)p . �6. Busemann–Petty type problems

The Busemann–Petty problem asks whether the implication

IK ⊂ IL �⇒ V (K) � V (L)

holds for arbitrary origin-symmetric K,L ∈ Kn0 . For 0 < p < 1, the Lp analogue of this question

asks: Does IpK ⊂ IpL for origin-symmetric K,L ∈ Kn0 imply V (K) � V (L)? We refer to this

question as the symmetric Lp Busemann–Petty problem. This was stated and solved in termsof polar L−p centroid bodies by Yaskin and Yaskina [40]. Their result shows that the answer ispositive if and only if n � 3. Since IpK ⊂ IpL is equivalent to I+p K ⊂ I+p L for origin-symmetricbodies K,L, the symmetric Lp Busemann–Petty problem asks whether

I+p K ⊂ I+p L �⇒ V (K) � V (L) (45)

holds for origin-symmetric bodies K,L ∈ Kn. If we allow the bodies in (45) to be arbitraryelements of Kn

0 , we call this question the nonsymmetric Lp Busemann–Petty problem.To each body K which is not origin-symmetric, one can construct bodies L such that the

desired implications for the original as well as the symmetric Lp Busemann–Petty problem fail.Our goal is to show that Lutwak’s connections on intersection bodies (which will be describedin detail below) also hold in the nonsymmetric Lp case. This proves in particular that thereare nonsymmetric bodies K for which (45) holds. Therefore we obtain a sufficient condition tocompare volumes of bodies which can be nonsymmetric.

Note that (45) is true for centered ellipsoids. This follows from (21) for E = F . Indeed,

V(I+p E

) = rn

I+pκ

2−n/pn V (E)n/p−1,

which immediately implies that (45) holds for ellipsoids.Lutwak’s first connection, as established in [25, Theorem 10.1], states that the answer to the

Busemann–Petty problem is affirmative if the body with smaller sections is an intersection body.The assumption of convexity of the involved bodies can be omitted in this case; the statementholds true for star bodies. The Lp analogue of this result is the following


Theorem 13. Let 0 < p < 1 and K,L ∈ Sn0 . If K is a nonsymmetric Lp intersection body, i.e.

contained in I+pSn, then

I+p K ⊂ I+p L,

implies

V (K) � V (L),

with equality only if K = L.

Proof. For a star body K with I+p K = K , the definition of dual Lp mixed volumes and Fubini’stheorem prove

V (K) = Vp(K,K) = Vp

(K, I+p K

), Vp(L,K) = Vp

(K, I+p L

).

Since

Vp

(K, I+p K

) = 1

n

∫Sn−1

ρ(K,u)n−p

(ρ(I+p K,u)

ρ(I+p L,u)

)p

ρ(I+p L,u

)pdu

� maxu∈Sn−1

(ρ(I+p K,u)

ρ(I+p L,u)

)p

Vp

(K, I+p L

),

we have

V (K)

Vp(L,K)� max

u∈Sn−1

(ρ(I+p K,u)

ρ(I+p L,u)

)p

. (46)

But I+p K ⊂ I+p L, so the claimed inequality for the volumes is an immediate consequence of(46) and (8). The equality case of the theorem follows from the equality case of the dual Lp

Minkowski inequality. �The next result is a negative counterpart of Theorem 13.

Theorem 14. Suppose we have an infinitely smooth star body L ∈ Sn0 which is not a nonsymmet-

ric Lp intersection body. Then there exists a star body K such that

ρ(I+p K, ·) < ρ

(I+p L, ·),

but

V (L) < V (K).


This is the Lp analogue of Lutwak’s second connection [25, Theorem 12.2] on intersectionbodies.

Proof. By Theorem 7 there exists a function f ∈ C∞(Sn−1) such that

ρ(L, ·)p = C+−pf.

Since L is not a nonsymmetric Lp intersection body, f must assume negative values. Thereforewe are able to choose a nonconstant function f ∈ C∞(Sn−1) such that

f (u) � 0, when f (u) < 0,

and

f (u) = 0, when f (u) � 0.

Choose another function f ∈ C∞(Sn−1) such that C+−pf = f . Now, since the origin is an interiorpoint of L, we can find a constant λ > 0 with

ρ(L, ·)n−p − λf > 0.

Define a star body Q by ρ(Q, ·)n−p := ρ(L, ·)n−p − λf (·). Then

ρ(I+p Q, ·)p = ρ

(I+p L, ·)p − λ

((n − p)�(1 − p)

)−1f .

Hence

ρ(I+p Q, ·)p � ρ

(I+p L, ·)p

, when f (u) < 0, (47)

and

ρ(I+p Q, ·)p = ρ

(I+p L, ·)p

, when f (u) � 0. (48)

The linearity properties of dual Lp mixed volumes and the self adjointness of C+−p yield

V (L) − Vp(Q,L) = 1

n

∫Sn−1

(ρ(L,u)n−p − ρ(Q,u)n−p

)ρ(L,u)p du

= 1

n

∫Sn−1

(ρ(L,u)n−p − ρ(Q,u)n−p

)C+−pf (u)du

= (n − p)�(1 − p)

n

∫Sn−1

(ρ(I+p L,u

)p − ρ(I+p Q,u

)p)f (u)du

< 0.


So from (8) we get

V (L) < V (Q).

Relations (47) and (48) show that I+p Q ⊂ I+p L. Set

ε :=(

1

2

(1 + V (L)

V (Q)

))1/n

.

Then ε < 1 and the body K := εQ has the desired properties. �Acknowledgment

The author is grateful to Monika Ludwig for many helpful discussions.

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